The non-Abelian counterpart of the photon Ward identity is a Slavnov-Taylor identity, derived from BRST symmetry. A scattering amplitude with all other external states physical vanishes when one external massless gauge boson polarization vector is replaced by its four-momentum. An amputated tensor with the remaining indices open need not be transverse: part d gives its explicit nonzero longitudinal remainder.
Consequently, amplitudes involving several unphysical covariant polarization vectors need not vanish. For example, contracting the result of part d with a timelike vector gives , nonzero in the explicit configuration there for . Even with other states physical, the pure-gauge vector is the relevant null combination; separate timelike and spatial longitudinal basis vectors are not themselves pure gauge, and their separate amplitudes need not be zero.
These auxiliary components do not define separately observable positive probabilities. Physical states are classes in BRST cohomology, and the S-matrix preserves that physical state space in this anomaly-free vectorlike gauge spectrum. There is no physical transition from a physical state to a nonphysical state. The incorrect stronger statement would be that every covariant component called “unphysical” has a zero matrix element.
For loop Feynman diagrams or unitarity cuts, summing covariant gauge indices with propagates those auxiliary components. In a Non-Abelian gauge theory, their cancellation requires the interacting Faddeev-Popov ghost fields as well as the gauge boson diagrams. Ghost contributions carry their Grassmann field loop signs and restore the Slavnov-Taylor identities and physical unitarity. One may instead use a formulation with only physical propagating polarization vectors, but one cannot simply discard the ghost diagrams while retaining the Feynman gauge rules from part a.
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